Radical proposals to overhaul the way English football is run could be backed by new legislation, following a withering parliamentary report criticising the failure to introduce new financial controls and increase the influence of fans.

The culture, media and sport select committee accuses the Football Association, the Premier League and the Football League of a "very disappointing" response to its report published in July 2011, which followed a wide-ranging inquiry into the governance of the game.

Unless there was "clear progress" in 12 months on those measures, the government should legislate "as soon as practicable", the committee says.

It wanted the FA to restructure its main board to assert its independence, overhaul the FA Council to make it more representative, introduce tough new rules on financial regulation, and increase the influence of supporters on how their clubs were run.

Hugh Robertson, the sports minister, said: "We welcome the report from the select committee which shows the will there is across parliament for football to modernise and change for the better.

"We have been clear that we want the football authorities to carry out the reforms they promised by the start of the 2013-14 season – most notably around improved governance and diverse representation at the FA, the development of a licensing system and greater financial transparency. If football does not deliver then we will look at bringing forward legislation."

John Whittingdale, chairman of the committee, said the FA and the two league bodies "simply don't address the fundamental problems" and that "much greater reform in football is needed to make the game inclusive, sustainable and driven from the grassroots".

In particular, he said, the committee was very disappointed with football's response to the call for a new licensing system run by the FA to bring some sanity to football's overheated finances.

Successive ministers have urged the FA to reform its structure but progress has been painfully slow. The comparatively modest reforms suggested by Lord Burns in 2005 have yet to be enforced in full. "We urge the authorities to be more radical and more urgent in addressing the problems faced by the game because of the weaknesses in its governance structure, both at FA and club level," said the committee.

The select committee argues that the FA should be an over-arching regulator for the game, able to dictate long-term strategy and direction. The Premier League, in contrast, argues it has always been an "association of interests" that exists to reflect the views of its "shareholders".

MPs make explicit that they believe the game to be too much in thrall to the influence of the Premier League. If anything, the committee concludes, football's proposals may lead to greater capture by the Premier League and "the regulated controlling the regulator".

"Altogether ... they fail to address the fundamental issue that the FA should exercise responsibility for all issues of major significance to the game through its main board and council," it said.

It also called for fundamental overhaul of the FA Council, which is supposed to be the game's parliament but is frequently criticised for a lack of diversity, and concrete proposals for greater fan influence at every club. The FA Council has 118 members, many of whom have served for more than 20 years while two-thirds are aged 64 or over.

MPs also criticised the football authorities and government for failing to bring forward "more detailed proposals for the involvement and consultation of supporters and supporters groups". It said government should set up an expert working group on the issue by the start of next season and resolve issues over how Supporters Direct, a body that provides help and advice to supporters' trusts, is funded.

The inquiry was set up in December 2010 after a series of controversies: Portsmouth became the first Premier League club to enter administration after passing through the hands of a series of owners; Liverpool became the subject of a complex high court legal battle; and fans protested against the debt-funded model of Manchester United's US owners.

Allied to renewed concerns over homegrown talent following an abject showing at the 2010 World Cup, the embarrassing failure of the bid to host the 2018 World Cup bid and longstanding concerns about the dysfunctional structure of the FA, Robertson was prompted to say that "football is the worst governed sport in this country, without a doubt".

The Premier League argues it has introduced a series of reforms in recent years designed to improve the transparency of club ownership and reduce financial recklessness. Concerned that its new £5.5bn broadcasting deal will fuel further rampant wage inflation, talks are continuing over introducing a version of the financial fair play breakeven rules imposed by Uefa. But the committee said the rules were unlikely to go far enough.

There is scepticism over whether the FA will be able to deliver even the reforms promised in the original joint submission. If the proposals made by football are not adopted in full by this summer, then the more radical changes lobbied for by the select committee and supporters' groups will be back on the agenda.

David Bernstein, the FA chairman who will be forced to stand down in the summer under rules preventing him from carrying on beyond his 70th birthday, has made some progress in successfully adding two non-executives to the main FA board - including its first ever woman in Heather Rabbatts - but has admitted progress is "slower than it might have been".

Clive Efford, the shadow sports minister, welcomed the report and said he shared the concern that "we have not moved forward far enough".

In a joint statement, the FA, the Football League and the Premier League said: "The football authorities continue to work towards the final approval and implementation of the governance reform proposals as outlined in February 2012. Significant headway has already been made on many of these proposed reforms, not least on sustainability and transparency.

"The remaining reform proposals are the subject of consultation within the game and we are confident that the necessary progress will be made."